Subgroups — C₂×#20272599 #27274176

Order 13568.

878 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810493	C₂	1	1	yes	maximal, characteristic
3	#20272599	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	—	2	1	no	—
6	C₂²×C₄⋊Dic₅₃	—	2	1	no	—
7	C₂²×C₄⋊Dic₅₃	—	2	1	no	—
8	C₂²×C₄⋊Dic₅₃	C₄	1	2	yes	—
9	C₂×C₄⋊#88322	—	2	2	no	—
10	C₂×C₄⋊#88322	—	2	1	no	—
11	C₂×C₄⋊#88322	C₄	1	4	yes	complemented
12	C₂×C₄⋊#88322	C₂²	1	2	yes	—
13	#16810493	—	2	1	no	—
14	#16810493	—	2	1	no	—
15	#16810493	C₄	1	2	yes	—
16	#16810493	C₂²	1	2	yes	—
17	C₂³×C₈×C₅₃	—	2	1	no	—
18	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
20	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
21	C₂×C₄⋊C₈×C₅₃	C₄	1	4	yes	complemented
22	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
23	C₂³×Dic₅₃	—	2	2	no	—
24	C₂²×#88322	—	2	4	no	—
25	C₂²×#88322	—	2	2	no	—
26	C₂²×#88322	—	4	2	no	—
27	C₂²×#88322	—	4	2	no	—
28	C₂×C₄⋊Dic₅₃	—	2	2	no	—
29	C₂×C₄⋊Dic₅₃	—	2	2	no	—
30	C₂×C₄⋊Dic₅₃	—	2	2	no	—
31	C₂×C₄⋊Dic₅₃	—	2	2	no	—
32	C₂×C₄⋊Dic₅₃	—	2	2	no	—
33	C₂×C₄⋊Dic₅₃	—	4	2	no	—
34	C₂×C₄⋊Dic₅₃	—	4	2	no	—
35	C₂×C₄⋊Dic₅₃	—	4	2	no	—
36	C₂×C₄⋊Dic₅₃	—	4	2	no	—
37	C₂×C₄⋊Dic₅₃	—	4	4	no	—
38	C₂×C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
39	C₄⋊#88322	—	2	4	no	—
40	C₄⋊#88322	—	2	2	no	—
41	C₄⋊#88322	C₂×C₄	1	4	yes	complemented
42	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	—	2	4	no	—
46	C₂²×C₈×C₅₃	—	2	2	no	—
47	C₂²×C₈×C₅₃	—	4	2	no	—
48	C₂²×C₈×C₅₃	—	4	2	no	—
49	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	D₄	1	2	yes	—
53	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
56	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₅₃	—	2	4	no	—
58	C₄⋊C₈×C₅₃	—	2	2	no	—
59	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
60	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	—	2	2	no	—
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₅₃	D₈	1	2	yes	—
73	C₂²×C₄×C₅₃	D₈	1	2	yes	—
74	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
75	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
76	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
77	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
78	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
79	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
80	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×Dic₅₃	—	2	4	no	—
84	C₂²×Dic₅₃	—	4	4	no	—
85	C₂²×Dic₅₃	—	4	4	no	—
86	C₂²×Dic₅₃	—	4	4	no	—
87	C₂×C₈×C₅₃	—	2	4	no	—
88	C₂×C₈×C₅₃	—	4	8	no	—
89	C₂×C₈×C₅₃	—	4	2	no	—
90	C₂×C₈×C₅₃	—	4	2	no	—
91	C₄²×C₅₃	C₄²	1	2	yes	—
92	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
93	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
94	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
95	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
96	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
97	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
101	C₄⋊Dic₅₃	—	4	4	no	—
102	C₄⋊Dic₅₃	—	4	4	no	—
103	C₄⋊Dic₅₃	—	4	4	no	—
104	C₄⋊Dic₅₃	—	4	2	no	—
105	C₄⋊Dic₅₃	—	4	2	no	—
106	C₄⋊Dic₅₃	—	4	2	no	—
107	C₄⋊Dic₅₃	—	4	2	no	—
108	C₄⋊Dic₅₃	—	4	4	no	—
109	C₂×#88322	—	2	4	no	—
110	C₂×#88322	—	4	8	no	—
111	C₂×#88322	—	4	2	no	—
112	C₂×#88322	—	4	2	no	—
113	C₂³×C₅₃	#95	1	2	yes	—
114	C₂³×C₅₃	#100	1	2	yes	—
115	C₂³×C₅₃	#101	1	2	yes	—
116	C₂³×C₅₃	#102	1	2	yes	—
117	C₂³×C₅₃	#102	1	2	yes	—
118	C₂³×C₅₃	#103	1	2	yes	—
119	C₂³×C₅₃	#103	1	2	yes	—
120	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	—	2	4	no	—
122	C₂×C₄×C₅₃	—	2	4	no	—
123	C₂×C₄×C₅₃	—	2	1	no	—
124	C₂×C₄×C₅₃	—	2	1	no	—
125	C₂×C₄×C₅₃	—	2	1	no	—
126	C₂×C₄×C₅₃	—	2	1	no	—
127	C₂×C₄×C₅₃	#95	1	2	yes	—
128	C₂×C₄×C₅₃	#95	1	2	yes	—
129	C₂×C₄×C₅₃	#95	1	2	yes	—
130	C₂×C₄×C₅₃	#102	1	2	yes	—
131	C₂×C₄×C₅₃	#102	1	4	yes	—
132	C₂×C₄×C₅₃	#102	1	2	yes	—
133	C₂×C₄×C₅₃	#102	1	4	yes	—
134	C₂×C₄×C₅₃	#103	1	2	yes	—
135	C₂×C₄×C₅₃	#103	1	4	yes	—
136	C₂×C₄×C₅₃	#103	1	2	yes	—
137	C₂×C₄×C₅₃	#103	1	4	yes	—
138	C₂×C₄×C₅₃	#106	1	4	yes	—
139	C₂×C₄×C₅₃	#106	1	4	yes	—
140	C₂×C₄×C₅₃	#107	1	4	yes	—
141	C₂×C₄×C₅₃	#107	1	4	yes	—
142	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
156	C₂×Dic₅₃	—	4	4	no	—
157	C₂×Dic₅₃	—	4	4	no	—
158	C₂×Dic₅₃	—	4	4	no	—
159	C₂×Dic₅₃	—	8	8	no	—
160	C₈×C₅₃	—	4	4	no	—
161	#88322	—	4	4	no	—
162	C₂×#1296	—	53	1	no	maximal, Sylow
163	C₂²×C₅₃	#331	1	4	yes	—
164	C₂²×C₅₃	#332	1	4	yes	—
165	C₂²×C₅₃	#332	1	4	yes	—
166	C₂²×C₅₃	#333	1	4	yes	—
167	C₂²×C₅₃	#340	1	4	yes	—
168	C₂²×C₅₃	#340	1	4	yes	—
169	C₂²×C₅₃	#343	1	4	yes	—
170	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
171	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
172	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
173	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
174	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
175	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
176	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
177	C₄×C₅₃	—	2	2	no	—
178	C₄×C₅₃	—	2	1	no	—
179	C₄×C₅₃	—	2	1	no	—
180	C₄×C₅₃	#340	1	4	yes	—
181	C₄×C₅₃	#340	1	4	yes	—
182	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
183	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
184	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
185	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
186	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
187	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
188	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
189	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
190	Dic₅₃	—	8	4	no	—
191	C₂²×C₄⋊C₈	—	53	1	no	—
192	C₂²×C₄⋊C₈	D₅₃	1	1	yes	characteristic
193	C₂×#384	—	53	1	no	—
194	#1296	—	53	4	no	—
195	C₂×C₅₃	#1296	1	8	yes	complemented
196	C₂×C₅₃	C₂×#331	1	1	yes	characteristic
197	C₂×C₅₃	C₂×#332	1	1	yes	characteristic
198	C₂×C₅₃	C₂×#332	1	1	yes	characteristic
199	C₂×C₅₃	C₂×#333	1	1	yes	characteristic
200	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
201	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
202	C₂×C₅₃	C₂×#343	1	1	yes	characteristic
203	C₂³×C₈	—	2	1	no	—
204	C₂³×C₈	—	106	1	no	—
205	C₂²×C₄⋊C₄	—	53	2	no	—
206	C₂²×C₄⋊C₄	—	106	1	no	—
207	C₂²×C₄⋊C₄	—	106	1	no	—
208	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
209	C₂×C₄⋊C₈	—	2	2	no	—
210	C₂×C₄⋊C₈	—	2	1	no	—
211	C₂×C₄⋊C₈	—	53	4	no	—
212	C₂×C₄⋊C₈	—	53	2	no	—
213	C₂×C₄⋊C₈	—	106	1	no	—
214	C₂×C₄⋊C₈	—	106	2	no	—
215	C₂×C₄⋊C₈	Dic₅₃	1	4	yes	complemented
216	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
217	#384	—	53	2	no	—
218	#384	—	53	2	no	—
219	#384	—	106	1	no	—
220	#384	—	106	1	no	—
221	C₅₃	C₂×#1296	1	1	yes	characteristic, complemented, Sylow
222	C₂³×C₄	—	106	2	no	—
223	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
224	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
225	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
226	C₂²×C₈	—	2	2	no	—
227	C₂²×C₈	—	2	4	no	—
228	C₂²×C₈	—	4	2	no	—
229	C₂²×C₈	—	4	2	no	—
230	C₂²×C₈	—	106	2	no	—
231	C₂²×C₈	—	106	4	no	—
232	C₂²×C₈	—	212	2	no	—
233	C₂²×C₈	—	212	2	no	—
234	C₂×C₄⋊C₄	—	53	4	no	—
235	C₂×C₄⋊C₄	—	106	2	no	—
236	C₂×C₄⋊C₄	—	106	2	no	—
237	C₂×C₄⋊C₄	—	106	2	no	—
238	C₂×C₄⋊C₄	—	106	2	no	—
239	C₂×C₄⋊C₄	—	106	2	no	—
240	C₂×C₄⋊C₄	—	212	4	no	—
241	C₂×C₄⋊C₄	—	212	2	no	—
242	C₂×C₄⋊C₄	—	212	2	no	—
243	C₂×C₄⋊C₄	—	212	2	no	—
244	C₂×C₄⋊C₄	—	212	2	no	—
245	C₂×C₄²	Dic₁₀₆	1	2	yes	—
246	C₂×C₄²	C₄×D₅₃	1	2	yes	—
247	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
248	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
249	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
250	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
251	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
252	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
253	C₄⋊C₈	—	2	2	no	—
254	C₄⋊C₈	—	2	4	no	—
255	C₄⋊C₈	—	53	4	no	—
256	C₄⋊C₈	—	106	2	no	—
257	C₄⋊C₈	—	106	4	no	—
258	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	complemented
259	C₂⁴	#11719921	1	1	yes	characteristic
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	106	4	no	—
263	C₂²×C₄	—	212	4	no	—
264	C₂²×C₄	—	212	4	no	—
265	C₂²×C₄	—	212	4	no	—
266	C₂²×C₄	#11719913	1	2	yes	—
267	C₂²×C₄	#11719913	1	2	yes	—
268	C₂²×C₄	D₄₂₄	1	2	yes	—
269	C₂²×C₄	Dic₂₁₂	1	2	yes	—
270	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
271	C₂²×C₄	#11719919	1	2	yes	—
272	C₂²×C₄	#11719919	1	2	yes	—
273	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
274	C₂²×C₄	#11719921	1	1	yes	characteristic
275	C₂²×C₄	#11719921	1	1	yes	characteristic
276	C₂²×C₄	#11719921	1	1	yes	characteristic
277	C₂²×C₄	#11719921	1	2	yes	—
278	C₂²×C₄	#11719922	1	2	yes	—
279	C₂²×C₄	#11719923	1	2	yes	—
280	C₂²×C₄	#11719924	1	2	yes	—
281	C₂²×C₄	#11719925	1	2	yes	—
282	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
283	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
284	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
285	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
286	C₂×C₈	—	2	4	no	—
287	C₂×C₈	—	4	8	no	—
288	C₂×C₈	—	4	2	no	—
289	C₂×C₈	—	4	2	no	—
290	C₂×C₈	—	106	4	no	—
291	C₂×C₈	—	212	8	no	—
292	C₂×C₈	—	212	2	no	—
293	C₂×C₈	—	212	2	no	—
294	C₄⋊C₄	—	212	4	no	—
295	C₄⋊C₄	—	212	4	no	—
296	C₄⋊C₄	—	212	4	no	—
297	C₄⋊C₄	—	212	4	no	—
298	C₄⋊C₄	—	212	2	no	—
299	C₄⋊C₄	—	212	2	no	—
300	C₄⋊C₄	—	212	2	no	—
301	C₄⋊C₄	—	212	2	no	—
302	C₄²	C₄×Dic₅₃	1	2	yes	—
303	C₄²	#11719919	1	2	yes	—
304	C₄²	#11719919	1	2	yes	—
305	C₄²	C₄⋊Dic₅₃	1	2	yes	—
306	C₄²	#11719921	1	2	yes	—
307	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
308	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
309	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
310	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
311	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
312	C₂³	#13133124	1	2	yes	—
313	C₂³	#13133133	1	2	yes	—
314	C₂³	#13133135	1	2	yes	—
315	C₂³	#13133137	1	2	yes	—
316	C₂³	#13133139	1	2	yes	—
317	C₂³	#13133140	1	2	yes	—
318	C₂³	#13133142	1	2	yes	—
319	C₂³	C₂×#11719921	1	1	yes	characteristic
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	4	no	—
325	C₂×C₄	—	2	4	no	—
326	C₂×C₄	—	424	8	no	—
327	C₂×C₄	—	212	4	no	—
328	C₂×C₄	—	212	4	no	—
329	C₂×C₄	—	212	4	no	—
330	C₂×C₄	#13133124	1	2	yes	—
331	C₂×C₄	#13133124	1	2	yes	—
332	C₂×C₄	#13133124	1	2	yes	—
333	C₂×C₄	#13133137	1	2	yes	—
334	C₂×C₄	#13133138	1	4	yes	—
335	C₂×C₄	#13133139	1	4	yes	—
336	C₂×C₄	#13133139	1	2	yes	—
337	C₂×C₄	#13133140	1	2	yes	—
338	C₂×C₄	#13133141	1	4	yes	—
339	C₂×C₄	#13133142	1	4	yes	—
340	C₂×C₄	#13133142	1	2	yes	—
341	C₂×C₄	#13133148	1	4	yes	—
342	C₂×C₄	#13133149	1	4	yes	—
343	C₂×C₄	#13133150	1	4	yes	—
344	C₂×C₄	#13133151	1	4	yes	—
345	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
346	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
347	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
348	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
349	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
350	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
351	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
352	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
353	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
354	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
355	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
356	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
357	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
358	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
359	C₈	—	4	4	no	—
360	C₈	—	212	4	no	—
361	C₂²	#16810366	1	4	yes	—
362	C₂²	#16810368	1	4	yes	—
363	C₂²	#16810370	1	4	yes	—
364	C₂²	#16810371	1	4	yes	—
365	C₂²	#16810385	1	4	yes	—
366	C₂²	#16810386	1	4	yes	—
367	C₂²	#16810391	1	4	yes	—
368	C₂²	C₂×#13133124	1	1	yes	characteristic
369	C₂²	C₂×#13133133	1	1	yes	characteristic
370	C₂²	C₂×#13133135	1	1	yes	characteristic
371	C₂²	C₂×#13133137	1	1	yes	characteristic
372	C₂²	C₂×#13133139	1	1	yes	characteristic
373	C₂²	C₂×#13133140	1	1	yes	characteristic
374	C₂²	C₂×#13133142	1	1	yes	characteristic
375	C₄	—	2	2	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	424	4	no	—
379	C₄	#16810385	1	4	yes	—
380	C₄	#16810386	1	4	yes	—
381	C₄	C₂×#13133138	1	1	yes	characteristic
382	C₄	C₂×#13133139	1	1	yes	characteristic
383	C₄	C₂×#13133141	1	1	yes	characteristic
384	C₄	C₂×#13133142	1	1	yes	characteristic
385	C₄	C₂×#13133148	1	1	yes	characteristic
386	C₄	C₂×#13133149	1	1	yes	characteristic
387	C₄	C₂×#13133150	1	1	yes	characteristic
388	C₄	C₂×#13133151	1	1	yes	characteristic
389	C₂	#20272599	1	8	yes	direct product factor
390	C₂	C₂×#16810366	1	1	yes	characteristic
391	C₂	C₂×#16810368	1	1	yes	characteristic
392	C₂	C₂×#16810370	1	1	yes	characteristic
393	C₂	C₂×#16810371	1	1	yes	characteristic
394	C₂	C₂×#16810385	1	1	yes	characteristic
395	C₂	C₂×#16810386	1	1	yes	characteristic
396	C₂	C₂×#16810391	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	#11719921
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂²×C₄⋊C₈×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂²×C₄⋊C₈	D₅₃
hypocenter	C₅₃	C₂×#1296
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #20272599	C₂	direct (N × H)
11. C₂×C₄⋊#88322	C₄	semidirect
21. C₂×C₄⋊C₈×C₅₃	C₄	semidirect
41. C₄⋊#88322	C₂×C₄	semidirect
59. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
195. C₂×C₅₃	#1296	semidirect
215. C₂×C₄⋊C₈	Dic₅₃	semidirect
221. C₅₃	C₂×#1296	semidirect
258. C₄⋊C₈	C₂×Dic₅₃	semidirect
389. C₂	#20272599	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄⋊#88322	2
C₂×#16810493	2
#20272599	2
C₂²×C₄⋊C₈×C₅₃	2
C₂×#1296	53

Subgroups — C2×#20272599 #27274176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272599 #27274176